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%I #26 Oct 11 2020 05:55:55
%S 1,6,12,23,40,68,113,186,304,495,804,1304,2113,3422,5540,8967,14512,
%T 23484,38001,61490,99496,160991,260492,421488,681985,1103478,1785468,
%U 2888951,4674424,7563380,12237809,19801194,32039008,51840207,83879220,135719432
%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C The titular polynomial is defined recursively by p(n,x)=x*p(n-1,x)+5*n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
%H Vincenzo Librandi, <a href="/A192754/b192754.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F Conjecture: G.f.: ( 1+4*x ) / ( (x-1)*(x^2+x-1) ), partial sums of A022095. a(n) = A000071(n+3)+4*A000071(n+2). - _R. J. Mathar_, May 04 2014
%F a(n) = 8*Fibonacci(n) + 3*Lucas(n) - 5. - _Greg Dresden_, Oct 10 2020
%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 1;
%t Table[Expand[p[n, x]], {n, 0, 7}]
%t reduce[{p1_, q_, s_, x_}] :=
%t FixedPoint[(s PolynomialQuotient @@ #1 +
%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t (* A192754 *)
%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t (* A192755 *)
%t LinearRecurrence[{2, 0, -1}, {1, 6, 12}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2012 *)
%Y Cf. A192744, A192232, A192755.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Jul 09 2011
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